In addition to conducting our analysis with biomass as the measurement of ecosystem function, in this document we report our results using Relative Yield Total (Loreau and Hector 2001; Wagg et al. 2019).
Mirroring the manuscript’s central analysis, our models for the across-treatment effect were encoded as: RelativeYieldTotal ~ -1 + Stage + Stage:Shannon, and the within-treatment effect was encoded as: RelativeYieldTotal ~ -1 + Richness:Stage + Richness:Stage:Shannon. All models successfully converged, with Rhat values of 1.0, and posterior predictive checks (PPC) were used to visually validate the model fits.
The relationship between Shannon diversity and Relative Yield Total was qualitatively similar to that of Shannon diversity for the majority of the models (5/6).
Within Grass3, positive within-treatment slopes estimating total biomass flip to negative when estimating Relative Yield Total. This leads the model to more closely reflect the other two grassland models. Because conspecific negative density dependence (CNDD) is present within the model, species’ monoculture biomasses are dramatically reduced in comparison to their biomasses within communities. Communities made of more competitive species will have lower Shannon diversity and higher Relative Yield Total, because there are few species present but they are released from the negative influence of CNDD. High diversity communities will generally have less total biomass per species, making each species’ contribution to Relative Yield Total smaller. Because the positive influence of CNDD saturates as individuals become surrounded by heterospecifics, this decrease in Relative Yield Total is not outweighed by any corresponding increases in the strength of CNDD. This creates negative within-treatment relationships between Shannon diversity and Relative Yield Total.
Considering the relationship between our measure of the internal coexistence processes within each model and the across-treatment effect of realized diversity on Relative Yield Total, we find that the aggregate patterns are nearly identical to those of total biomass.
Considering the relationship between our measure of the internal coexistence processes within each model and the within-treatment effect of realized diversity on Relative Yield Total, we find that the aggregate patterns are nearly identical to those of total biomass.
This section of the document describes the statistical models’ validation, using Shannon diversity as the focal biodiversity metric and Relative Yield Total as the focal ecosystem function.
Important terms:
Stage: With seed rain, Without seed rainNinitial: Planted species richnessClark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 642)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain -14.82 5.37 -25.37 -4.21 1.00 2003 1860
## StageWithoutseedrain -27.46 5.76 -38.81 -16.00 1.00 1718 1844
## StageWithseedrain:Shannon 17.32 2.08 13.30 21.33 1.00 2038 1924
## StageWithoutseedrain:Shannon 26.56 2.60 21.46 31.66 1.00 1712 2074
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 22.58 0.65 21.36 23.89 1.00 2624 2106
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2128202 0.02514011 0.163788 0.2614227
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 25.80 22.30 -17.44 69.12 1.01 6006 3054
## Ninitial4:StageWithseedrain 93.73 20.16 54.85 133.85 1.00 4772 2716
## Ninitial8:StageWithseedrain 127.34 16.23 95.09 158.47 1.00 5285 2956
## Ninitial16:StageWithseedrain 216.62 14.21 188.05 244.77 1.00 6024 2579
## Ninitial32:StageWithseedrain 250.36 16.71 216.95 283.24 1.00 5654 2640
## Ninitial2:StageWithoutseedrain 3.83 8.61 -13.21 20.40 1.00 5441 2667
## Ninitial4:StageWithoutseedrain 38.47 12.33 14.02 61.99 1.00 5499 2860
## Ninitial8:StageWithoutseedrain 66.81 12.34 42.85 90.99 1.00 5254 3246
## Ninitial16:StageWithoutseedrain 163.64 15.55 133.39 193.91 1.00 5492 2757
## Ninitial32:StageWithoutseedrain 174.27 19.85 135.11 212.55 1.00 5103 2850
## Ninitial2:StageWithseedrain:Shannon -12.15 13.88 -39.15 14.78 1.01 5984 3101
## Ninitial4:StageWithseedrain:Shannon -36.26 9.23 -54.71 -18.44 1.00 4808 2652
## Ninitial8:StageWithseedrain:Shannon -39.54 6.13 -51.40 -27.44 1.00 5328 2977
## Ninitial16:StageWithseedrain:Shannon -59.82 4.82 -69.35 -50.19 1.00 5963 2604
## Ninitial32:StageWithseedrain:Shannon -61.76 5.27 -72.15 -51.12 1.00 5585 2587
## Ninitial2:StageWithoutseedrain:Shannon 0.99 5.79 -10.07 12.52 1.00 5587 2762
## Ninitial4:StageWithoutseedrain:Shannon -12.78 6.32 -25.05 -0.24 1.00 5471 2913
## Ninitial8:StageWithoutseedrain:Shannon -19.85 5.46 -30.58 -9.30 1.00 5282 3039
## Ninitial16:StageWithoutseedrain:Shannon -49.75 6.41 -62.10 -37.26 1.00 5541 2939
## Ninitial32:StageWithoutseedrain:Shannon -43.65 7.57 -58.18 -28.76 1.00 5060 2962
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 12.72 0.36 12.02 13.42 1.00 7473 2989
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.7499603 0.008676023 0.7318249 0.7657269
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 642)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 3.11 1.33 0.54 5.77 1.00 1770 1782
## StageWithoutseedrain -5.47 1.43 -8.26 -2.66 1.00 1861 1908
## StageWithseedrain:Shannon 6.26 0.43 5.39 7.11 1.00 1782 1857
## StageWithoutseedrain:Shannon 12.65 0.58 11.50 13.80 1.00 1816 1834
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 6.95 0.20 6.58 7.35 1.00 2804 2405
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5269179 0.01839451 0.489196 0.5598625
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 178.46 19.56 139.60 217.41 1.00 5531 2739
## Ninitial4:StageWithseedrain 218.83 20.20 178.90 259.29 1.00 5266 2855
## Ninitial8:StageWithseedrain 186.97 18.12 152.57 221.97 1.00 4432 2768
## Ninitial16:StageWithseedrain 138.56 28.38 83.44 194.73 1.00 4827 2868
## Ninitial32:StageWithseedrain 102.51 34.09 37.72 171.06 1.00 5255 2911
## Ninitial2:StageWithoutseedrain 23.47 12.90 -2.24 49.25 1.00 4773 2937
## Ninitial4:StageWithoutseedrain -24.26 7.60 -39.35 -9.37 1.00 4865 2737
## Ninitial8:StageWithoutseedrain -14.80 6.04 -26.96 -2.93 1.00 4361 2615
## Ninitial16:StageWithoutseedrain 9.26 6.20 -2.91 21.34 1.00 5185 2860
## Ninitial32:StageWithoutseedrain 36.63 8.10 20.18 52.49 1.00 5072 2708
## Ninitial2:StageWithseedrain:Shannon -101.24 11.85 -125.05 -77.65 1.00 5525 2778
## Ninitial4:StageWithseedrain:Shannon -87.77 8.76 -105.25 -70.44 1.00 5270 2922
## Ninitial8:StageWithseedrain:Shannon -56.06 6.25 -68.10 -44.01 1.00 4450 2735
## Ninitial16:StageWithseedrain:Shannon -31.73 8.00 -47.66 -16.13 1.01 4816 2867
## Ninitial32:StageWithseedrain:Shannon -17.71 8.13 -34.05 -2.34 1.00 5270 2867
## Ninitial2:StageWithoutseedrain:Shannon -7.70 7.97 -23.57 8.22 1.00 4813 2825
## Ninitial4:StageWithoutseedrain:Shannon 22.44 4.13 14.42 30.50 1.00 4861 2575
## Ninitial8:StageWithoutseedrain:Shannon 19.15 2.74 13.79 24.62 1.00 4352 2636
## Ninitial16:StageWithoutseedrain:Shannon 8.01 2.27 3.54 12.50 1.00 5160 2757
## Ninitial32:StageWithoutseedrain:Shannon -0.60 2.42 -5.31 4.40 1.00 5078 2728
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 5.10 0.14 4.82 5.39 1.00 6722 3015
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.7489654 0.008953145 0.7303685 0.7649965
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 642)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain -10.03 0.83 -11.71 -8.45 1.00 2181 1807
## StageWithoutseedrain -10.05 0.83 -11.71 -8.43 1.00 1899 1651
## StageWithseedrain:Shannon 4.10 0.29 3.56 4.68 1.00 2205 1657
## StageWithoutseedrain:Shannon 4.32 0.30 3.72 4.95 1.00 1841 1837
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 3.85 0.11 3.63 4.06 1.00 2806 2038
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.3965623 0.02392655 0.3482008 0.4415797
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 37.34 17.42 3.83 70.87 1.00 3770 3156
## Ninitial4:StageWithseedrain 75.41 11.98 51.92 99.07 1.00 4000 3132
## Ninitial8:StageWithseedrain 57.71 11.92 34.59 81.15 1.00 4076 3284
## Ninitial16:StageWithseedrain 56.38 10.70 34.86 77.25 1.00 4055 3157
## Ninitial32:StageWithseedrain 42.56 14.14 15.37 69.38 1.00 3978 2998
## Ninitial2:StageWithoutseedrain -4.76 3.04 -10.85 1.01 1.00 3612 2514
## Ninitial4:StageWithoutseedrain 19.80 5.14 9.46 29.73 1.00 3863 3057
## Ninitial8:StageWithoutseedrain 20.36 5.41 9.55 30.76 1.00 4331 3068
## Ninitial16:StageWithoutseedrain 50.84 10.07 31.14 70.67 1.00 4388 2891
## Ninitial32:StageWithoutseedrain 58.53 14.92 29.42 88.31 1.00 3617 2640
## Ninitial2:StageWithseedrain:Shannon -24.33 10.44 -44.38 -4.27 1.00 3752 3113
## Ninitial4:StageWithseedrain:Shannon -33.38 5.25 -43.76 -23.06 1.00 3997 3122
## Ninitial8:StageWithseedrain:Shannon -19.97 4.21 -28.21 -11.77 1.00 4065 3313
## Ninitial16:StageWithseedrain:Shannon -15.75 3.17 -21.96 -9.40 1.00 4037 3112
## Ninitial32:StageWithseedrain:Shannon -9.46 3.71 -16.51 -2.39 1.00 3987 3057
## Ninitial2:StageWithoutseedrain:Shannon 0.84 1.88 -2.80 4.59 1.00 3606 2512
## Ninitial4:StageWithoutseedrain:Shannon -9.69 2.40 -14.32 -4.86 1.00 3878 3023
## Ninitial8:StageWithoutseedrain:Shannon -7.39 2.06 -11.32 -3.30 1.00 4348 3035
## Ninitial16:StageWithoutseedrain:Shannon -15.37 3.16 -21.62 -9.17 1.00 4397 2889
## Ninitial32:StageWithoutseedrain:Shannon -14.56 4.24 -23.09 -6.34 1.00 3642 2373
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 3.14 0.09 2.98 3.33 1.00 7302 2604
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.6015697 0.01540604 0.5709656 0.6298231
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 642)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 13.11 2.75 7.65 18.34 1.00 1991 1991
## StageWithoutseedrain 15.61 2.92 10.01 21.37 1.00 1902 2193
## StageWithseedrain:Shannon 7.66 1.55 4.73 10.74 1.00 1978 1905
## StageWithoutseedrain:Shannon 6.31 2.04 2.20 10.28 1.00 1940 2212
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 15.33 0.43 14.49 16.19 1.00 2852 2361
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.05731091 0.01686307 0.02755178 0.09281329
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 44.73 7.27 30.32 58.80 1.00 4528 2782
## Ninitial4:StageWithseedrain 61.57 6.11 49.94 73.80 1.00 4655 2536
## Ninitial8:StageWithseedrain 66.74 7.00 53.04 80.60 1.00 4205 3151
## Ninitial16:StageWithseedrain 56.76 8.82 39.32 74.04 1.00 4858 3188
## Ninitial32:StageWithseedrain 28.88 13.56 2.10 55.62 1.00 4324 2842
## Ninitial2:StageWithoutseedrain 31.55 9.25 13.32 49.28 1.00 3814 2994
## Ninitial4:StageWithoutseedrain 48.19 5.69 37.36 59.66 1.00 3869 3238
## Ninitial8:StageWithoutseedrain 69.04 6.04 57.21 80.98 1.00 4878 2736
## Ninitial16:StageWithoutseedrain 32.04 6.12 20.00 43.66 1.00 4728 2810
## Ninitial32:StageWithoutseedrain 20.55 5.56 9.84 31.43 1.00 4202 3149
## Ninitial2:StageWithseedrain:Shannon -29.62 6.05 -41.36 -17.57 1.00 4526 2679
## Ninitial4:StageWithseedrain:Shannon -31.45 4.58 -40.56 -22.59 1.00 4558 2524
## Ninitial8:StageWithseedrain:Shannon -23.97 4.49 -32.64 -15.08 1.00 4080 3202
## Ninitial16:StageWithseedrain:Shannon -11.71 4.42 -20.30 -3.04 1.00 4886 3024
## Ninitial32:StageWithseedrain:Shannon 2.80 5.49 -8.06 13.74 1.00 4345 2935
## Ninitial2:StageWithoutseedrain:Shannon -21.18 8.68 -37.90 -3.91 1.00 3925 3032
## Ninitial4:StageWithoutseedrain:Shannon -24.93 4.63 -34.38 -16.04 1.00 3905 3079
## Ninitial8:StageWithoutseedrain:Shannon -30.72 4.55 -39.67 -21.62 1.00 4822 2657
## Ninitial16:StageWithoutseedrain:Shannon -0.21 3.95 -7.82 7.57 1.00 4784 2806
## Ninitial32:StageWithoutseedrain:Shannon 6.73 3.09 0.73 12.73 1.00 4330 2846
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 11.25 0.32 10.64 11.90 1.00 8368 2653
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5039633 0.01964115 0.4633785 0.539916
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 642)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 1.61 1.03 -0.48 3.56 1.00 1941 1867
## StageWithoutseedrain 0.31 1.24 -2.15 2.72 1.00 1809 1985
## StageWithseedrain:Shannon -0.21 0.36 -0.90 0.50 1.00 1899 1888
## StageWithoutseedrain:Shannon -4.80 0.64 -6.01 -3.55 1.00 1835 2070
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 5.84 0.16 5.54 6.16 1.00 2878 2639
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4437236 0.02170308 0.3990911 0.4846449
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 17.29 6.82 3.75 30.33 1.00 3913 2796
## Ninitial4:StageWithseedrain 46.38 7.95 30.97 62.16 1.00 5153 3161
## Ninitial8:StageWithseedrain 29.79 11.68 7.22 52.35 1.00 3392 2711
## Ninitial16:StageWithseedrain 45.52 19.25 8.08 82.69 1.00 3526 3086
## Ninitial32:StageWithseedrain 83.10 27.42 29.91 136.66 1.00 3620 2871
## Ninitial2:StageWithoutseedrain -3.77 3.68 -10.85 3.40 1.00 3881 2776
## Ninitial4:StageWithoutseedrain -10.70 3.53 -17.48 -3.65 1.00 3874 2990
## Ninitial8:StageWithoutseedrain -6.03 3.87 -13.54 1.63 1.00 3901 2896
## Ninitial16:StageWithoutseedrain -14.19 3.33 -20.69 -7.70 1.00 4292 2898
## Ninitial32:StageWithoutseedrain -13.61 4.84 -23.14 -4.15 1.00 4599 2960
## Ninitial2:StageWithseedrain:Shannon -11.24 4.48 -19.87 -2.38 1.00 3944 2839
## Ninitial4:StageWithseedrain:Shannon -21.64 3.76 -29.06 -14.35 1.00 5213 3188
## Ninitial8:StageWithseedrain:Shannon -10.27 4.36 -18.63 -1.88 1.00 3311 2774
## Ninitial16:StageWithseedrain:Shannon -13.38 5.75 -24.48 -2.07 1.00 3520 3192
## Ninitial32:StageWithseedrain:Shannon -20.87 6.97 -34.50 -7.30 1.00 3613 2916
## Ninitial2:StageWithoutseedrain:Shannon 0.36 2.72 -5.03 5.65 1.00 3840 2805
## Ninitial4:StageWithoutseedrain:Shannon 2.57 2.22 -1.81 6.84 1.00 3891 2992
## Ninitial8:StageWithoutseedrain:Shannon -1.54 1.93 -5.33 2.27 1.00 3928 2789
## Ninitial16:StageWithoutseedrain:Shannon 1.11 1.57 -1.89 4.14 1.00 4219 2858
## Ninitial32:StageWithoutseedrain:Shannon 0.33 1.95 -3.52 4.15 1.00 4557 3218
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 5.36 0.15 5.08 5.66 1.00 7919 3241
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5408732 0.01758599 0.5049973 0.5747371
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 642)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 0.20 1.93 -3.49 4.08 1.00 2168 2104
## StageWithoutseedrain -4.07 2.38 -8.75 0.60 1.01 2192 2206
## StageWithseedrain:Shannon 0.79 0.76 -0.68 2.23 1.00 2118 2105
## StageWithoutseedrain:Shannon -0.19 1.42 -2.99 2.60 1.00 2167 2107
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 10.07 0.27 9.57 10.61 1.00 2663 2675
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.09981672 0.02094118 0.06084377 0.1425852
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain -5.07 7.67 -20.16 9.85 1.00 3809 3031
## Ninitial4:StageWithseedrain -15.28 10.52 -36.05 5.10 1.00 3659 3120
## Ninitial8:StageWithseedrain -34.97 12.91 -60.10 -8.67 1.00 3770 2949
## Ninitial16:StageWithseedrain -34.01 18.29 -69.55 2.04 1.00 3269 2941
## Ninitial32:StageWithseedrain -72.47 23.74 -116.97 -25.30 1.00 3158 2920
## Ninitial2:StageWithoutseedrain -2.04 5.07 -11.66 8.01 1.00 3978 2893
## Ninitial4:StageWithoutseedrain -20.20 5.23 -30.26 -9.54 1.00 4210 2693
## Ninitial8:StageWithoutseedrain -21.93 5.41 -32.33 -11.26 1.00 4112 2804
## Ninitial16:StageWithoutseedrain -36.46 7.57 -51.36 -21.61 1.00 3949 2974
## Ninitial32:StageWithoutseedrain -44.48 11.38 -67.33 -22.32 1.00 3581 2963
## Ninitial2:StageWithseedrain:Shannon 5.30 5.27 -5.08 15.57 1.00 3885 3061
## Ninitial4:StageWithseedrain:Shannon 9.10 5.50 -1.46 19.88 1.00 3671 2979
## Ninitial8:StageWithseedrain:Shannon 15.27 5.32 4.58 25.81 1.00 3812 2848
## Ninitial16:StageWithseedrain:Shannon 12.10 6.19 0.05 24.20 1.00 3303 2937
## Ninitial32:StageWithseedrain:Shannon 21.65 6.85 8.11 34.60 1.00 3075 2932
## Ninitial2:StageWithoutseedrain:Shannon 3.27 4.03 -4.48 11.28 1.00 4024 2669
## Ninitial4:StageWithoutseedrain:Shannon 12.43 3.57 5.20 19.33 1.00 4156 2969
## Ninitial8:StageWithoutseedrain:Shannon 10.14 3.23 3.77 16.43 1.00 4152 2907
## Ninitial16:StageWithoutseedrain:Shannon 15.78 4.05 7.89 23.70 1.00 3922 3122
## Ninitial32:StageWithoutseedrain:Shannon 18.05 5.68 7.15 29.30 1.00 3609 3091
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.33 0.27 8.82 9.86 1.00 7696 3081
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2566906 0.02466921 0.2071976 0.3037057
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.